During the first 5 or 6 weeks of the quarter, we will continue our study of properties of varieties (Part II of Algebraic Geometry: A First Course by Joe Harris), at a slightly faster pace than during winter quarter. Topics will be chosen from the following list.

• Continuation of our study of dimension (Lectures 11 and 12)
• Smoothness and tangent spaces (Lecture 15 in the text)
• Degree (and some specific examples) (Lectures 18 and 19 in the text)
• Singular points and tangent cones (Lecture 21 in the text)

If time permits, we'll briefly discuss Hilbert polynomials (Lecture 15), since the degree and dimension of a projective variety are both related to the Hilbert polynomial of its homogeneous coordinate ring.

The last 4 or 5 weeks of the course will be a very brief introduction to the theory of schemes, especially those parts of scheme theory that are most closely related to questions about algebraic varieties. The following topics are fairly representative:

• The sheaf of regular functions on a variety
• Affine schemes
  • Topology
  • Structure sheaf
  • Examples
• Affine schemes vis à vis affine varieties
  • Generic points
  • Examples of nilpotent elements in the structure sheaf
• Projective schemes
• An algebraic proof of the algebraic case of Sard's theorem

As in Math 8203 and Math 8204, the required work in the course will consist of solving exercises from the text and writing up the solutions.

Spring Quarter office hours

M,F	1:30 to 3:30
W	1:30 to 2:30

Back to the main syllabus

Last updated Wednesday, April 8, 1998.

Joel Roberts
351 Vincent Hall
625-1076
e-mail: roberts@math.umn.edu
http://www.math.umn.edu/~roberts